Complex transformations and mappings

7. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { ( 1 + \mathrm { i } ) z + 2 ( 1 - \mathrm { i } ) } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation maps points on the imaginary axis in the \(z\)-plane onto a line in the \(w\)-plane.

1. Find an equation for this line. The transformation maps points on the real axis in the \(z\)-plane onto a circle in the \(w\)-plane.
2. Find the centre and radius of this circle.

1. A complex number \(z\) is represented by the point \(P\) in an Argand diagram.

Given that $$| z - 2 i | = | z - 3 |$$

1. sketch the locus of \(P\). You do not need to find the coordinates of any intercepts. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { \mathrm { i } z } { z - 2 \mathrm { i } } \quad z \neq 2 \mathrm { i }$$ Given that \(T\) maps \(| z - 2 i | = | z - 3 |\) to a circle \(C\) in the \(w\)-plane,
2. find the equation of \(C\), giving your answer in the form $$| w - ( p + q \mathrm { i } ) | = r$$ where \(p , q\) and \(r\) are real numbers to be determined.

1. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { z } { z + 4 \mathrm { i } } \quad z \neq - 4 \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the circle \(C\) Determine

1. a Cartesian equation of \(C\)
2. the centre and radius of \(C\)

1. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\) is given by

$$w = \frac { z + 1 } { z - 3 } \quad z \neq 3$$ The straight line in the \(z\)-plane with equation \(y = 4 x\) is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane.

1. Show that \(C\) has equation $$3 u ^ { 2 } + 3 v ^ { 2 } - 2 u + v + k = 0$$ where \(k\) is a constant to be determined.
2. Hence determine
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\)

1. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { z - \mathrm { i } } { z + 1 } \quad z \neq - 1$$ Given that \(T\) maps the imaginary axis in the \(z\)-plane to the circle \(C\) in the \(w\)-plane, determine (i) the coordinates of the centre of \(C\) (ii) the radius of \(C\)

8. In the Argand diagram the point \(P\) represents the complex number \(z\). Given that arg \(\left( \frac { z - 2 \mathrm { i } } { z + 2 } \right) = \frac { \pi } { 2 }\),

1. sketch the locus of \(P\),
2. deduce the value of \(| \mathrm { z } + 1 - \mathrm { i } |\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is defined by $$w = \frac { 2 ( 1 + \mathrm { i } ) } { z + 2 } , \quad z \neq - 2$$
3. Show that the locus of \(P\) in the \(z\)-plane is mapped to part of a straight line in the \(w\)-plane, and show this in an Argand diagram.
   (6)(Total 12 marks)

9. (a) The point \(P\) represents a complex number \(z\) in an Argand diagram. Given that $$| z - 2 i | = 2 | z + i |$$

1. find a cartesian equation for the locus of \(P\), simplifying your answer.
2. sketch the locus of \(P\).
   (b) A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is a translation \(- 7 + 11\) i followed by an enlargement with centre the origin and scale factor 3 . Write down the transformation \(T\) in the form $$w = a z + b , \quad a , b \in \mathbb { C }$$

1. (i) (a) On the same Argand diagram sketch the loci given by the following equations.

$$| z - 1 | = 1 , \quad , , \arg ( z + 1 ) = \frac { \pi } { 12 } , \quad , \arg ( z + 1 ) = \frac { \pi } { 2 }$$ (b) Shade on your diagram the region for which $$| z - 1 | \leq 1 \quad \text { and } \quad \frac { \pi } { 12 } \leq \arg ( z + 1 ) \leq \frac { \pi } { 2 }$$ (ii) (a) Show that the transformation \(\quad w = \frac { z - 1 } { z } , \quad z \neq 0\), $$\text { maps } | z - 1 | = 1 \text { in the } \boldsymbol { z } \text {-plane onto } | w | = | w - 1 | \text { in the } \boldsymbol { w } \text {-plane. }$$ The region \(| z - 1 | \leq 1\) in the \(z\)-plane is mapped onto the region \(T\) in the \(w\)-plane.
(b) Shade the region \(T\) on an Argand diagram.

9. A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$| z - 3 \mathrm { i } | = 3$$

1. sketch the locus of \(P\).
2. Find the complex number \(z\) which satisfies both \(| z - 3 i | = 3\) and \(\arg ( z - 3 i ) = \frac { 3 } { 4 } \pi\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 2 \mathrm { i } } { z }$$
3. Show that \(T\) maps \(| z - 3 i | = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line.
   (5)(Total 11 marks)

6. A complex number \(z\) is represented by the point \(P\) in the Argand diagram.

1. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
2. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
3. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.

1. The point \(P\) represents the complex number \(z\) on an Argand diagram, where

$$| z - \mathrm { i } | = 2$$ The locus of \(P\) as \(z\) varies is the curve \(C\).

1. Find a cartesian equation of \(C\).
2. Sketch the curve \(C\). A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + \mathrm { i } } { 3 + \mathrm { i } z } , \quad z \neq 3 \mathrm { i }$$ The point \(Q\) is mapped by \(T\) onto the point \(R\). Given that \(R\) lies on the real axis,
3. show that \(Q\) lies on \(C\).

8. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.

1. Given that \(| z | = 1\), sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 7 \mathrm { i } } { z - 2 \mathrm { i } }$$
2. Show that \(T\) maps \(| z | = 1\) onto a circle in the \(w\)-plane.
3. Show that this circle has its centre at \(w = - 5\) and find its radius.

3. A transformation maps points from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\). The transformation is given by $$w = \frac { ( 2 + \mathrm { i } ) z + 4 } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation maps the imaginary axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Determine a Cartesian equation of \(l\), giving your answer in the form \(a u + b v + c = 0\) where \(a , b\) and \(c\) are integers to be found.
(6)

4. A complex number \(z\) is represented by the point \(P\) in an Argand diagram. Given that $$| z + i | = 1$$

1. sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$
2. Given that \(T\) maps \(| z + i | = 1\) to a circle \(C\) in the \(w\)-plane, find a cartesian equation of \(C\).

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively in Argand diagrams and $$w = z ^ { 2 } - 1$$

1. Show that \(v = 2 x y\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 3 x\). Find the equation of the locus of \(Q\).

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively, in Argand diagrams, and \(w = 1 - z ^ { 2 }\).

1. Find expressions for \(u\) and \(v\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 4 x\). Find the equation of the locus of \(Q\).
3. Find the perpendicular distance of the point corresponding to \(z = 2 + 5 \mathrm { i }\) in the \(( u , v )\)-plane, from the locus of \(Q\).

1. The complex numbers \(z\) and \(w\) are represented, respectively, by the points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and

$$w = \frac { z } { 1 - z }$$

1. Show that \(v = \frac { y } { ( 1 - x ) ^ { 2 } + y ^ { 2 } }\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 1 - x\). Find and simplify the equation of the locus of \(Q\).

7. The complex numbers \(z\) and \(w\) are represented, respectively, by points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and $$w = z ( 1 + z )$$

1. Show that $$v = y ( 1 + 2 x )$$ and find an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = x + 1\). Find the Cartesian equation of the locus of \(Q\), giving your answer in the form \(v = a u ^ { 2 } + b u\), where \(a\) and \(b\) are constants whose values are to be determined.

1. A transformation from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } \quad z \neq - \mathrm { i }$$

1. Show that the circle \(C\) with equation \(| z + \mathrm { i } | = 1\) in the \(z\)-plane is mapped to a circle \(D\) in the \(w\)-plane, giving a Cartesian equation for \(D\).
2. Sketch \(C\) and \(D\) on Argand diagrams.

1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { 1 - 3 z } { z + 2 i } \quad z \neq - 2 i$$ The circle with equation \(| z + \mathrm { i } | = 3\) is mapped by \(T\) onto the circle \(C\).

1. Show that the equation for \(C\) can be written as $$3 | w + 3 | = | 1 + ( 3 - w ) \mathrm { i } |$$
2. Hence find
   1. a Cartesian equation for \(C\),
   2. the centre and radius of \(C\).

1. 1. A circle \(C\) in the complex plane is defined by the locus of points satisfying

$$| z - 3 i | = 2 | z |$$

1. Determine a Cartesian equation for \(C\), giving your answer in simplest form.
2. On an Argand diagram, shade the region defined by $$\{ z \in \mathbb { C } : | z - 3 \mathrm { i } | > 2 | z | \}$$ (ii) The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = z ^ { 3 }$$
3. Describe the geometric effect of \(T\). The region \(R\) in the \(z\)-plane is given by $$\left\{ z \in \mathbb { C } : 0 < \arg z < \frac { \pi } { 4 } \right\}$$
4. On a different Argand diagram, sketch the image of \(R\) under \(T\).

1. A transformation from the \(z\)-plane to the \(w\)-plane is given by

$$w = z ^ { 2 }$$

1. Show that the line with equation \(\operatorname { Im } ( z ) = 1\) in the \(z\)-plane is mapped to a parabola in the \(w\)-plane, giving an equation for this parabola.
2. Sketch the parabola on an Argand diagram.

6. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { Z } { Z + \mathrm { i } } , \quad Z \neq - \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the curve \(C\).

1. Show that \(C\) is a circle and find its centre and radius. The region \(| z | < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
2. Shade the region \(R\) on an Argand diagram.